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Homework 6 Math 352, Fall 2011 Instructions: Solve both of these problems. Your solutions must be written in LATEX. Due Date: Friday, November 4 1. Let C be the cylinder x2 + y 2 = 1, and let f : C → R3 be the function f (x, y, z) = (x cos z, y cos z, sin z). Recall that the vectors t1 = (−y, x, 0) and t2 = (0, 0, 1) are a basis for the tangent space to C at each point. (a) Find the 3 × 2 matrix for Df with respect to the basis {t1 , t2 }. (b) Compute the set of all critical points for f . Describe this set geometrically. (c) Prove that the image of f is precisely the unit sphere S 2 . (d) Find the preimages under f of the points (0, 0, 1), (0, 0, −1), and (1, 0, 0). Describe these preimages geometrically. 2. Use the method of Lagrange multipliers to solve the following problems: (a) Find the four critical points for the function f (x, y, z) = xy + z 2 on the cylinder x2 + y 2 = 1. (b) Find the six critical points for the function g(x, y, z) = xz + yz on the unit sphere.